Let $C = (3, 5)$ be the midpoint of $AB,$ where $A = (1, 8)$ and $B = (x, y).$ What is the value of $xy$?
Answer: Clearly $C$ must be the midpoint of $AB$ if $C$ is the closest point to both $A$ and $B.$ Using the midpoint formula, we see that: $$\left(\frac{1 + x}{2}, \frac{8 + y}{2}\right) = \left(3, 5\right).$$Therefore, $\frac{1 + x}{2} = 3$, so $x = 5.$ Meanwhile, $\frac{8 + y}{2} = 5$, so $y = 2$. Our answer is $xy = \boxed{10}.$